How would you completely factor: a(x) =8x^3 - 12x^2 + 18x -27 without knowing any of the roots?

Answer 1

Use a technique called factoring by grouping. You should read up on this as it is a very useful technique

Will make into 2 groups

8x^3 -12x^. +. 18x -27. Then factor each group separately

4x^2(2x -3). + 9( 2x-3). Both these pairs have a common factor of 2x-3

So can rewrite as (4x^2 +9)( 2x-3) and there you have it

Answer 2

the ratio of the x^3 to the x term is 8:18, or 4:9 in simplest form.

the ratio of the x^2 to the constant term is -12:-27 or 4:9 in simplest form.

When you have this pattern, it tells you what the factors are:

(a x + b) (c x^2 + d)

a is the factor for the first ratio (2)

b is the factor for the second ratio (-3)

c and d are in the ratio:

(2 x - 3) ( 4 x^2 + 9 )

This is a popular pattern for cubics that get factored in Algebra 2.